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Question
Write the following function in the simplest form:
`tan^(-1) x/(sqrt(a^2 - x^2))`, |x| < a
Solution
`tan^(-1) x/(sqrt(a^2 - x^2))`
Put `x = asin theta`
`=> x/a = sin theta`
`=> theta = sin^(-1) (x/a)`
`:. tan^(-1) x/sqrt(a^2 - x^2) `
`= tan^(-1) ((a sin theta)/(sqrt(a^2 - a^2 sin^2 theta)))`
`= tan^(-1) ((asin theta)/(asqrt(1-sin^2 theta))) `
`= tan^(-1) ((asin theta)/(acos theta))`
`= tan^(-1) (tan theta)`
`= theta `
`= sin^(-1) x/a`
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